We consider representations of solutions to equations of the form u_t = Au(t,x) + g(u(t,x)), u(0,x) = f(x), on S,
for a locally compact space S, where (A,D_A) is a densely defined linear operator on C(S) for the uniform norm satisfying the conditions of the Hille-Yosida theorem to generate a continuous semigroup of postive, linear contractions on C(S); here C(S) is the space of all continuous functions on S when S is compact, or C_0(S) when S is locally compact, but not compact. We will consider the three cases of g(u) = \pm ru, and g(u) = ru(u-1), with r non-negative; \pm r being two cases. The interplay between semigroup and martingale theories will be described in connection with these equations. Most of the talk describes well-known results and will hopefully be self-contained for the uninitiated, but it will end with a new result that grew out of joint work with William Felder on the drift paradox in mathematical biology/ecology. (For those concerned that it may not be an analysis talk, the proof does use integration by parts !).